Modified gravity

Modified gravity changes the Einstein-Hilbert action or its field equations in order to produce gravitational dynamics different from general relativity.

A scalar-tensor theory uses a metric tensor and an additional scalar field to mediate gravity. A scalar multiplier of the Ricci scalar makes the curvature coupling depend on that field. The Brans-Dicke theory is a standard example; the choice of scalar potential and kinetic coupling distinguishes different members of the class.

With the curvature sign convention in which $\delta R_{bc}={\nabla }_c\delta {\Gamma }^a{}_{ba}-{\nabla }_a\delta {\Gamma }^a{}_{bc}$, the inverse-metric variation of $\int \sqrt{-g}\Phi R$ is $\int \sqrt{-g}{E}_{ab}\delta g^{ab}$ plus a boundary term, for a scalar multiplier held fixed in that variation. The Palatini identity expresses the curvature variation as divergences of $\delta \Gamma$. Two integrations by parts move the derivatives onto $\Phi$, yielding the displayed terms. For constant $\Phi$ they vanish. The opposite curvature convention instead has $E_{ab}=\Phi G_{ab}+g_{ab}\square \Phi -{\nabla }_a{\nabla }_b\Phi$.

The Brans-Dicke theory adds a dynamical scalar field $\Phi$ to a metric tensor theory of gravity, with constant dimensionless coupling $\omega$. In a $(+---)$ signature and the reversed-curvature convention used here, the displayed Lagrangian is supplemented by matter and integrated against the metric tensor volume density with prefactor $1/(16\pi G)$. Its scalar multiplier of the Ricci scalar lets the curvature coupling vary in spacetime. Both the metric tensor and scalar are varied; the scalar equation of Brans-Dicke theory follows by eliminating curvature between the scalar equation and the trace of the metric tensor equation. The original construction is described in Brans and Dicke's 1961 article.

Assume four spacetime dimensions, $\Phi \ne 0$, constant $\omega$, no scalar potential and matter independent of $\Phi$. In the stated action convention, scalar variation gives $\Phi R+\omega {\Phi }^{-1}(\nabla \Phi {)}^2-2\omega \square \Phi =0$. The trace of the metric tensor equation gives $\Phi R+\omega {\Phi }^{-1}(\nabla \Phi {)}^2+3\square \Phi =8\pi GT$. Their subtraction proves the displayed wave equation with $T=g^{ab}{T}_{ab}$. The degenerate coupling of Brans-Dicke theory needs separate treatment before dividing by its coefficient.

At $\omega =-3/2$, the combined scalar equation of Brans-Dicke theory becomes the matter consistency condition $T=0$. It does not determine $\square \Phi$ by division, since the wave coefficient has vanished. The original metric tensor and scalar Euler-Lagrange equations must be retained. Introducing scalar-dependent matter or a scalar potential changes the consistency condition.

An f(R) theory replaces the Ricci scalar in the gravitational action by a smooth function $f(R)$. Its metric field equation is

In the mostly-minus convention with vacuum gravitational action proportional to $\int \sqrt{-g}f(R)$ and $f^{\prime }<0$, the displayed conformal transformation turns its vacuum equation into ${\tilde{G}}_{\mu \nu }=T_{\mu \nu }(\varphi )$, with potential $V=(R{f}^{\prime }-f)/(2f^{\prime 2})$ and the stress-energy tensor of a canonical scalar field. The conformal Ricci formula cancels the second derivatives of $f^{\prime }$ against the original field equation. Its remaining derivative tensor is ${\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -{\tilde{g}}_{\mu \nu }(\tilde{\nabla }\varphi {)}^2/2$. Expressing $V$ as a function of $\varphi$ requires a locally invertible $R\mapsto f^{\prime }(R)$. When $f^{\prime \prime }=0$, this scalar-tensor parametrization degenerates and introduces no new propagating scalar degree of freedom.

Every four-dimensional Einstein metric with $R_{ab}=\Lambda g_{ab}$ has constant Ricci scalar $R=4\Lambda$. The vacuum metric f(R) gravity tensor reduces to $(\Lambda f^{\prime }(4\Lambda )-f(4\Lambda )/2)g_{ab}$, giving the displayed necessary and sufficient condition for a fixed cosmological constant. Both function and derivative may vanish, so division by $f^{\prime }$ is unnecessary. Requiring this for all real $\Lambda$ and a smooth $f$ instead yields $f(R)=C{R}^2$.

For a metric action $S_g=A\int \sqrt{-g}f(R)d^{4}x$ and the usual stress-energy tensor convention, covariant metric variation gives $A{E}_{ab}=T_{ab}/2$, where $E_{ab}=f^{\prime }{R}_{ab}-f{g}_{ab}/2+(g_{ab}\square -{\nabla }_a{\nabla }_b)f^{\prime }$. With $A=1/(16\pi G)$ this becomes $E_{ab}=8\pi G{T}_{ab}$; with $A=1$ it becomes $E_{ab}=T_{ab}/2$. The chosen normalization does not alter the vacuum equation.

If $f(0)=0$, $f^{\prime }(0)=1$, and $f^{\prime \prime }(0)=0$, then $f(R)=R+O(R^3)$ near zero. The field equations linearized about Minkowski spacetime are therefore the Linearized Einstein equations; deviations first appear beyond linear order.